Directional harmonic theory: a computational Gestalt model to account for illusory contour and vertex formation

Visual illusions and perceptual grouping phenomena offer an invaluable tool for probing the computational mechanism of low-level visual processing. Some illusions, like the Kanizsa figure, reveal illusory contours that form edges collinear with the inducing stimulus. This kind of illusory contour has been modeled by neural network models by way of cells equipped with elongated spatial receptive fields designed to detect and complete the collinear alignment. There are, however, other illusory groupings which are not so easy to account for in neural network terms. The Ehrenstein illusion exhibits an illusory contour that forms a contour orthogonal to the stimulus instead of collinear with it. Other perceptual grouping effects reveal illusory contours that exhibit a sharp corner or vertex, and still others take the form of vertices defined by the intersection of three, four, or more illusory contours that meet at a point. A direct extension of the collinear completion models to account for these phenomena tends towards a combinatorial explosion, because it would suggest cells with specialized receptive fields configured to perform each of those completion types, each of which would have to be replicated at every location and every orientation across the visual field. These phenomena therefore challenge the adequacy of the neural network approach to account for these diverse perceptual phenomena. I have proposed elsewhere an alternative paradigm of neurocomputation in the harmonic resonance theory (Lehar 1999, see website), whereby pattern recognition and completion are performed by spatial standing waves across the neural substrate. The standing waves perform a computational function analogous to that of the spatial receptive fields of the neural network approach, except that, unlike that paradigm, a single resonance mechanism performs a function equivalent to a whole array of spatial receptive fields of different spatial configurations and of different orientations, and thereby avoids the combinatorial explosion inherent in the older paradigm. The present paper presents the directional harmonic model, a more specific development of the harmonic resonance theory, designed to account for specific perceptual grouping phenomena. Computer simulations of the directional harmonic model show that it can account for collinear contours as observed in the Kanizsa figure, orthogonal contours as seen in the Ehrenstein illusion, and a number of illusory vertex percepts composed of two, three, or more illusory contours that meet in a variety of configurations.     

What goes up may come down: perceptual process and knowledge access in the organization of complex visual patterns by young infants

The relationship between perceptual categorization and organization processes in 3‐ to 4‐month‐old infants was explored. The question was whether an invariant part abstracted during category learning could interfere with Gestalt organizational processes. Experiment 1 showed that the infants could parse a circle in accord with good continuation from visual patterns consisting of a circle and a complex polygon. In Experiments 2 and 3, however, this parsing was interfered with by a prior category familiarization experience in which infants were presented with visual patterns consisting of a pacman shape and a complex polygon. Part 1 of Experiments 2 and 3 showed that the infants recognized the pacman as familiar, and Part 2 demonstrated that the representation of the pacman blocked the subsequent parsing of the circle. The results suggest that a cognitive system of flexible feature creation can override organizational principles with which a perceptual system may come pre‐equipped.     

The interplay between perceptual organization and categorization in the representation of complex visual patterns by young infants

The relation between perceptual organization and categorization processes in 3- and 4-month-olds was explored. The question was whether an invariant part abstracted during category learning could interfere with Gestalt organizational processes. A 2003 study by Quinn and Schyns had reported that an initial category familiarization experience in which infants were presented with visual patterns consisting of a pacman shape and a complex polygon could interfere with infants' subsequent good continuation-based parsing of a circle from visual patterns consisting of a circle and a complex polygon. However, an alternative noninterference explanation for the results was possible because the pacman had been presented with greater frequency and duration than had the circle. The current study repeated Quinn and Schyns's procedure but provided an equivalent number of familiarization trials and duration of study time for the infants to process the pacman during initial familiarization and the circle during subsequent familiarization. The results replicated the previous findings of Quinn and Schyns. The data are consistent with the interference account and suggest that a cognitive system of adaptable feature creation can take precedence over organizational principles with which a perceptual system comes pre-equipped.     

Perception: A HyperCard stack for demonstrating visual perceptual phenomena

Perception is a HyperCard stack that allows users to explore visual illusions and other perceptual phenomena on the Apple Macintosh. The stack contains over 20 demonstrations of intersecting line illusions, size and shape illusions, subjective contours, color assimilation, and so forth. As a presentation tool for classroom or laboratory demonstrations, Perception offers three unique features for displaying visual phenomena: (1) the capability to “dissolve” the inducing elements of an illusion in order to show the objective state of affairs, (2) the ability to quickly reverse the inducing elements of an illusion and therefore the effects of the distortion, and (3) animation of the various components of an illusion to produce continuous distortions in-real time. These features are illustrated with use of the Orbison, Titchener, Hering, and Wundt illusions. Use of the stack reveals two interesting and unanticipated findings: (1) an apparent size distortion in the central square of the Orbison illusion as it moves back and forth across the background of concentric rings, and (2) perceptual aftereffects that arise when the inducing elements of the Titchener, Hering, or Wundt illusion is dissolved.     

Properties of long-range illusory contours produced by offset-arcs

Researchers have used several different types of illusory contours to investigate properties of human perception. One rarely used illusory contour is a combination of the abutting grating and Kanizsa illusions. We call this the offset-arcs illusion and provide an empirical investigation of the illusion. Through a series of four experiments, using different methods of measurement, we show that changes to the phase of the abutting-grating part of the inducing stimulus can dramatically change the perceived strength and clarity of the long-range illusory contour. The easy manipulation of illusion strength should make the offset-arcs illusion applicable to a wide range of studies that utilize long-range illusory contours. The lack of a brightness component to the illusion should allow the offset-arcs illusion to help separate perceptual grouping from surface brightness effects that are often confounded in other illusory contours.     

The Relationship Between Visual Illusion and Aesthetic Preference – an Attempt to Unify Experimental Phenomenology and Empirical Aesthetics

Experimental phenomenology has demonstrated that perception is much richer than stimulus. As is seen in color perception, one and the same stimulus provides more than several modes of appearance or perceptual dimensions. Similarly, there are various perceptual dimensions in form perception. Even a simple geometrical figure inducing visual illusion gives not only perceptual impressions of size, shape, slant, depth, and orientation, but also affective or aesthetic impressions. The present study reviews our experimental phenomenological work on visual illusion and experimental aesthetics, and examines how aesthetic preference is influenced by stimulus factors determining visual illusions including anomalous surface and transparency as well as geometrical illusion. Along with line figures producing geometrical illusions, illusory surface figures inducing neon color spreading and transparency effects were used as test patterns. Participants made both of psychophysical judgments and of aesthetic judgments for the same test pattern. Both of geometrical illusions and aesthetic preferences were found to change similarly as a function of stimulus variables such as the number of filling lines and the size ratio of the inner and outer figural components. Also, following specific stimulus variables such as lightness contrast ratio and spatial interval between inducing figural elements (so called ``packmen''), strong effects of color spreading and transparency were accompanied with strong preferences. It seems that the paradigm to investigate aesthetic phenomena along with perceptual dimensions is useful to bridge the gap between experimental phenomenology and experimental aesthetics.     

Subjective contours and the Poggendorff illusion

Poggendorff illusions were generated by real edges, subjective contours, and various control patterns. Using both magnitude estimation and reproduction measures of illusion strength, it was found that subjective contours produced a reliable Poggendorff illusion. This clarifies previous reports which could not demonstrate a subjective contour-based illusion.     

Attentional processing and the subjective contour illusion

Leading explanations of the subjective contour illusion can be classified as being either "bottom-up" or "top-down." Bottom-up explanations assert that peripheral, physiological mechanisms often associated with the perception of real contours also account for subjective contour (SC) perception. In contrast, top-down explanations posit a more central locus of SC perception and are formulated on a molar, psychological level. A major aspect of bottom-up perceptual processing is that it is largely automatic. On the other hand, top-down processing implies a greater role for selective attention. In an effort to distinguish between bottom-up and top-down accounts of SC perception, the present investigation used a dual-task paradigm to test the relative attentional demands of real contour perception versus SC perception. In the primary task, subjects made speeded same-different discriminations of either paired SC forms or their real contour analogues. Half the subjects performed this primary task in conjunction with a six-digit short-term memory load secondary task. If subjective forms indeed impose a greater limited-capacity processing load than real forms, then the need to share processing capacity with a secondary task was expected to produce a greater increment in reaction time (RT) for subjective relative to real forms. The results indicated that the expected enhanced RT increment for subjective relative to real forms with the addition of a concurrent memory load was limited to same trials. This result implies that the nature of response indicators must be considered in assessing capacity requirements with the sort of dual-task paradigm used in the present investigation. Nevertheless, the fact that the increment in same RT with the addition of a concurrent memory load was greater for subjective relative to real forms accords with expectations derived from the notion that the perception of SCs is more attention demanding than that of real contours. If the interpretation of the present study is correct, then a comprehensive theory of SC perception will most likely be formulated within the top-down perspective of conceptually driven visual information processing.     

Effects of contour proximity and lightness on Delboeuf illusions created by circumscribed letters

32 observers judged the size of a letter, either an "A" or an "S," which was surrounded by a circle. Both letters were overestimated, but larger surrounding circles reduced the illusion. Decreasing the lightness contrast of the surrounding circle relative to the central letter diminished the illusion. The results suggest that, like the Delboeuf illusion, these circumscribed letters illusions are produced by interactions among size-coding neurons.     

Photopic visual phantom illusion: its common and unique characteristics as a completion effect

An illusion similar to the stationary visual phantom illusion presented earlier by Gyoba (1983, Vision Research 23 205-211) is reported. This illusion is visible in photopic vision and we have tentatively named it the 'photopic phantom illusion'. A typical example of this illusion is a white and light-gray square-wave grating occluded by a black region. In this figure, a phantom grating running across the occluder with clear contours but less contrast, is seen. The critical spatial frequencies of photopic phantoms have been measured and compared with those of scotopic phantoms that have been reported previously, revealing a great resemblance between them. We discuss the characteristics of this illusion in terms of transparency, stereoscopic viewing, and perceptual completion.     

Making features similar: comparison processes affect perception

In this paper, the constructive nature of comparison processes (both similarity- and differenceoriented judgments) is examined through their effects on visual perception. Previous research has shown that comparison processes enhance the tendency to interpret ambiguous objects in the light of the unambiguous objects with which they are compared (Medin, Goldstone, & Gentner, 1993). In the present paper, it is argued that comparison processes affect not only the interpretation of objects, but also their perception. In addition, it is argued that the perceptual effects of similarity-oriented comparison processes differ from those of difference-oriented comparison processes. Accordingly, it is demonstrated that when estimation of an object’s size is preceded by similarity-oriented comparisons, the Ebbinghaus illusion practically disappears, whereas prior difference-oriented comparisons tend to enhance the illusion.     

Perception of symmetry and repetition within and across visual shapes: Part-descriptions and object-based attention

In five experiments, we investigated the detection of symmetry (i.e., translation plus reflection) or repetition (i.e., translation alone) between two vertical jagged contours. The complexity of the two contours was manipulated, as was their figure-ground assignment; the two contours either belonged to a common object "inside" them, to two separate objects "outside" them, or to two separate objects each to the right of one contour. Replicating Baylis and Driver (1994), symmetry judgements were unaffected by contour complexity when made within a single shape, implying a parallel process operating efficiently across contour discontinuities. However, all the other conditions yielded substantially slower judgements as complexity increased, suggesting either effortful point-by-point comparisons, or a highly inefficient parallel process. In agreement with Baylis and Driver (1995a), symmetry perception was harder when figure-ground assignment turned convexities along one contour into concavities along the other contour; and likewise for repetition detection. However, even when convex parts matched between the two contours, judgements were still affected by complexity unless they belonged to a common object. This supports Baylis and Driver's (1993) proposal that effortless comparisons for the layout of multiple convex parts can only be made within single perceptual objects.     

The illusion of transparency and chromatic subjective contours

Subjective contours can be produced that include an illusion of edge and an extension of color throughout the area of the illusion. The phenomenological appearance is of a transparent colored shape in front of the background. Two explanations of this illusion are proposed. The first is that there is an assimilation of color analogous to brightness assimilation. The second is a variant of the stratification of depth theory of subjective contours. In it, the pattern elements lead to the illusion of a surface in front of the pattern elements. We thus predicted that an illusion of transparency would enhance the subjective contour, Metelli’s model of transparency was used to quantify our prediction, and it was found that the possibility of transparency was a powerful predictor of the chromatic subjective contour.     

Hierarchical organisation in perception of orientation

According to Rock [1990, in The Legacy of Solomon Asch (Hillsdale, NJ: Lawrence Erlbaum Associates)], hierarchical organisation of perception describes cases in which the orientation of an object is affected by the immediately surrounding elements in the visual field. Various experiments were performed to study the hierarchical organisation of orientation perception. In most of them the rod-and-frame-illusion (RFI: change of the apparent vertical measured on a central rod surrounded by a tilted frame) was measured in the presence/absence of a second inner frame. The first three experiments showed that, when the inner frame is vertical, the direction and size of the illusion are consistent with expectancies based on the hierarchical organisation hypothesis. An analysis of published and unpublished data collected on a large number of subjects showed that orientational hierarchical effects are independent from the absolute size of the RFI. In experiments 4 to 7 we examined the perceptual conditions of the inner stimulus (enclosure, orientation, and presence of luminance borders) critical for obtaining a hierarchical organisation effect. Although an inner vertical square was effective in reducing the illusion (experiment 3), an inner circle enclosing the rod was ineffective (experiment 4). This indicates that definite orientation is necessary to modulate the illusion. However, orientational information provided by a vertical or horizontal rectangle presented near the rod, but not enclosing it, did not modulate the RFI (experiment 5). This suggests that the presence of a figure with oriented contours enclosing the rod is critical. In experiments 6 and 7 we studied whether the presence of luminance borders is important or whether the inner upright square might be effective also if made of subjective contours. When the subjective contour figure was salient and the observers perceived it clearly, its effectiveness in modulating the RFI was comparable to that observed with luminance borders.     

Stereokinetic Anomalous Contours: Demonstrations

Collinearity or correspondence between the contours of the inducing figure to allow `contour continuation' or `figure completion' were, according to G. Kanizsa, the necessary conditions for producing anomalous surfaces or contours. Since Kanizsa's early work various hypotheses have been advanced to explain the phenomenon, but very few examples of anomalous contours that do not satisfy the above conditions have been reported. When two small white discs (1 cm in diameter) are set on a larger black disc in slow rotation, the two discs, after some observation time, will appear as the extremities of a rigid cylinder displaced in depth. The surface of the cylinder, under dim illumination, appears as a whitish transparent surface. However, when the two discs are substituted by a circle and a semicircle of the same size, a clear anomalous contour appears to form the cylinder, even under clear light conditions and when the colours are reversed; i.e., black circles on white disc. The anomalous contours are not apparent when the configuration is stationary. I will demonstrate how the anomalous contours of a stereokinetic cylinder can be obtained even without the “interruption” of the lines in the semicircle. The relationship between the anomalous contours of the stereokinetic cylinder and the vitreous transparency of the surface of the cylinder formed by the two small discs above mentioned, will be discussed as well as their relation to the general theories of anomalous surfaces.     

Application of the rubber hand illusion paradigm: comparison between upper and lower limbs

The “rubber hand illusion (RHI)” is a perceptual illusion, which allows the integration of artificial limbs into the body representation of a person by means of combined visual and tactile stimulation. The illusion has been frequently replicated but always concerning the upper limbs. The present study verified an analog illusion that can be called the “rubber foot illusion” (RFI). In a conjoint experiment using both a rubber hand and a rubber foot, brushstrokes were applied to the respective real and rubber limb placed alongside the real one. However, only the artificial limb’s handling was visible. The brushstrokes were given either synchronously, with a delay of ±0.5 s, or without tactile stimulation of the real limb. Questionnaire data and the proprioceptive drift towards the rubber limb (determined by calling on the subjects to show where they locate their unseen limb) defined the illusion strength. Results revealed that the illusion was induced in both limbs with comparable strength, but only in the synchronous condition.     

Background stripes affect apparent speed of rotation

A gray line that rotated about its own center against a stationary background of vertical stripes appeared to double in perceptual speed as it rotated through the vertical position and thus momentarily aligned with the background. Four factors may contribute to this speed-up: (i) landmarks, in which the tip of the moving vertical line moves horizontally across the maximum number of stationary stripes; (ii) orientation repulsion of the moving line by the vertical stripes, which may distort the line's perceived position and hence its perceived speed; (iii) the orientation of an induced brightness pattern along the line; and (iv) the motion of the induced brightness pattern, which moves physically most rapidly along the line when the line is near vertical. To test these possibilities, an annulus display provided landmarks but no intersections, and this almost abolished the effect. A rotating-slit display provided an oriented, moving pattern that mimicked the induced brightness but had no landmarks, and this increased the effect. We conclude that the motion, but not the orientation, of the intersections [option (iv)] was responsible for the illusion. The fact that this motion along the length of the line affected the perceived speed of the line orthogonal to its own length indicates a failure on the part of the visual system to fully decouple tangential from radial motion.     

The tactual horizontal-vertical illusion depends on radial motion of the entire arm

Wesought to clarify the causes of the tactual horizontal-vertical illusion, where vertical lines are overestimated as compared with horizontals in Land inverted-T figures. Experiment 1 did not use L or inverted-T figures, but examined continuous or bisected horizontal and vertical lines. It was expected that bisected lines would be perceived as shorter than continuous lines, as in the inverted-T figure in the horizontal-vertical illusion. Experiment 1 showed that the illusion could not be explained solely by bisection, since illusory effects were similar for continuous and bisected vertical and horizontal lines. Experiments 2 and 3 showed that the illusory effects were dependent upon stimulus size and scanning strategy. Overestimation of the vertical was minimal or absent for the smallest patterns, where it was proposed that stimuli were explored by finger movement, with flexion at the wrist. Larger stimuli induce whole-arm motions, and illusory effects were found in conditions requiring radial arm motion. The illusion was weakened or eliminated in Experiment 4 when subjects were forced to examine stimuli with finger-and-hand motion alone, that is, their elbows were kept down on the table surface, and they were prevented from making radial arm motions. Whole-arm motion damaged performance and induced perceptual error. The experiments support the hypothesis that overestimation of the vertical in the tactual horizontal-vertical illusion derives from radial scanning by the entire arm.     

Model for a three-dimensional optical illusion

A homogeneous coordinate system is used to describe the transformation from a real three-dimensional stimulus to an illusory three-dimensional perceptual object. The model comprises a series of transformations of which one acts as an illusion operator. The illusion operator is specified by a single parameter whose value determines whether the real or the illusory object is perceived. An experiment to test one prediction derived from the model was performed. The results confirm the prediction.     

3-D shape perception

In this paper, we analyze and test three theories of 3-D shape perception: (1) Helmholtzian theory, which assumes that perception of the shape of an object involves reconstructing Euclidean structure of the object (up to size scaling) from the object’s retinal image after taking into account the object’s orientation relative to the observer, (2) Gibsonian theory, which assumes that shape perception involves invariants (projective or affine) computed directly from the object’s retinal image, and (3) perspective invariants theory, which assumes that shape perception involves a new kind of invariants of perspective transformation. Predictions of these three theories were tested in four experiments. In the first experiment, we showed that reliable discrimination between a perspective and nonperspective image of a random polygon is possible even when information only about the contour of the image is present. In the second experiment, we showed that discrimination performance did not benefit from the presence of a textured surface, providing information about the 3-D orientation of the polygon, and that the subjects could not reliably discriminate between the 3-D orientation of the textured surface and that of a shape. In the third experiment, we compared discrimination for solid shapes that either had flat contours (cuboids) or did not have visible flat contours (cylinders). The discrimination was very reliable in the case of cuboids but not in the case of cylinders. In the fourth experiment, we tested the effectiveness of planar motion in perception of distances and showed that the discrimination threshold was large and similar to thresholds when other cues to 3-D orientation were used. All these results support perspective invariants as a model of 3-D shape perception.